The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation

D. G. Aronson; J. L. Vazquez

Annales de l'I.H.P. Analyse non linéaire (1987)

  • Volume: 4, Issue: 3, page 203-230
  • ISSN: 0294-1449

How to cite

top

Aronson, D. G., and Vazquez, J. L.. "The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation." Annales de l'I.H.P. Analyse non linéaire 4.3 (1987): 203-230. <http://eudml.org/doc/78129>.

@article{Aronson1987,
author = {Aronson, D. G., Vazquez, J. L.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {porous medium flow; Hamilton-Jacobi equation; viscosity solutions interfaces; finite speed; propagation of disturbances},
language = {eng},
number = {3},
pages = {203-230},
publisher = {Gauthier-Villars},
title = {The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation},
url = {http://eudml.org/doc/78129},
volume = {4},
year = {1987},
}

TY - JOUR
AU - Aronson, D. G.
AU - Vazquez, J. L.
TI - The porous medium equation as a finite-speed approximation to a Hamilton-Jacobi equation
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1987
PB - Gauthier-Villars
VL - 4
IS - 3
SP - 203
EP - 230
LA - eng
KW - porous medium flow; Hamilton-Jacobi equation; viscosity solutions interfaces; finite speed; propagation of disturbances
UR - http://eudml.org/doc/78129
ER -

References

top
  1. [A1] D.G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math., t. 17, 1969, p. 461-467. Zbl0187.03401MR247303
  2. [A 2] D.G. Aronson, Regularity properties of flows through porous media: the interface, Arch. Rational Mech. Anal., t. 37, 1970, p. 1-10. Zbl0202.37901MR255996
  3. [AB] D.G. Aronson, Ph. Bénilan, Régularité des solutions de l'équation des milieux poreux dans R", C. R. Acad. Sci. Paris, t. 288, 1979, p. 103-105. Zbl0397.35034MR524760
  4. [AC] D.G. Aronson, L.A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc., t. 280, 1983, p. 351-366. Zbl0556.76084MR712265
  5. [ACK] D.G. Aronson, L.A. Caffarelli, S. Kamin, How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal., t. 14, 1983, p. 639-658. Zbl0542.76119MR704481
  6. [ACV] D.G. Aronson, L.A. Caffarelli, J.L. Vazquez, Interfaces with a corner-point in one-dimensional porous medium flow, Comm. Pure Applied Math., t. 38, 1985, p. 375-404. Zbl0544.35058MR792397
  7. [BC 1] Ph. Bénilan, M.G. Crandall, The continuous dependence on φ of the solutions of ut - Δφ(u) = 0, Indiana Univ. Math. J., t. 30, 1981, p. 161-177. Zbl0482.35012MR604277
  8. [BC2] Ph. Bénilan, M.G. Crandall, Regularizing effects of homogeneous evolution equations, Contributions to Analysis and Geometry, Supplement to American J. Math., D. N. Clark et al. eds., Baltimore, 1981, p. 23-39. Zbl0556.35067MR648452
  9. [BCP] Ph. Bénilan, M.G. Crandall, M. Pierre, Solutions of the porous medium equation in R'' under optimal conditions on initial values, Indiana Univ. Math. J., t. 33, 1984, p. 51-87. Zbl0552.35045MR726106
  10. [BV] Ph. Bénilan, J.L. Vazquez, Concavity of solutions of the porous medium equation, to appear in Trans. Amer. Math. Soc. Zbl0628.76092MR869400
  11. [B] S.H. Benton, The Hamilton-Jacobi equation : a global approach, Academic Press, New York, 1977. Zbl0418.49001MR442431
  12. [CF] L.A. Caffarelli, A. Friedman, Regularity of the free-boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math., t. 101, 1979, p. 1193-1218. Zbl0439.76084MR548877
  13. [CEL] M.G. Crandall, L.C. Evans, P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., t. 282, 1984, p. 487-502. Zbl0543.35011MR732102
  14. [CL] M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., t. 277, 1983, p. 1-42. Zbl0599.35024MR690039
  15. [D] C.M. Dafermos, Characteristics in hyperbolic conservation laws in Nonlinear Analysis and Mechanics, Vol. I, ed. R. J. Knops, London, Pitman, 1977, p. 1-58. Zbl0373.35048MR481581
  16. [DK] B.E. Dahlberg, C.E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Diff. Equations, t. 9, 1984, p. 409-437. Zbl0547.35057MR741215
  17. [G] B.H. Gilding, Hölder continuity of solutions of parabolic equations, J. London Math. Soc., t. 13, 1976, p. 103-106. Zbl0319.35045MR399658
  18. [GJ] J.L. Graveleau, P. Jamet, A finite-different approach to some degenerate non-linear parabolic equations, SIAM J. Appl. Math., t. 20, 1971, p. 199-223. Zbl0226.65065MR290600
  19. [HP] M.A. Herrero, M. Pierre, The Cauchy problem for ut = Δum when 0 &lt; m &lt; 1, Trans. Amer. Math. Soc., t. 290, 1985, p. 1-14. 
  20. [H] E. Hopf, The partial differential equation ut + uux = μuxx. Comm. Pure Applied Math., t. 3, 1950, p. 201-230. Zbl0039.10403MR47234
  21. [K] S. Kamin, The asymptotic behaviour of the solution of the filtration equation, Israel J. Math., t. 14, 1973, p. 76-78. Zbl0254.35054MR315292
  22. [KC] W.L. Kath, D.S. Cohen, Waiting time behaviour in a nonlinear diffusion equation. Studies in Applied Maths., t. 67, 1982, p. 79-105. Zbl0499.76078MR670736
  23. [Kn] B. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc., t. 234, 1977, p. 381-415. Zbl0365.35030MR492856
  24. [La] P.D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Applied Math., t. 10, 1957, p. 537-566. Zbl0081.08803MR93653
  25. [Li] P.L. Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math. # 69, Pitman, Boston, 1982. Zbl0497.35001MR667669
  26. [LSV] P.L. Lions, P.E. Souganidis, J.L. Vazquez, In preparation. 
  27. [LP] T.-P. Liu, M. Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Diff. Equations, t. 51, 1984, p. 419-441. Zbl0545.35057MR735207
  28. [O] O.A. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Translations, t. 26, (2), 1983, p. 95-172. Doklady Akad. Nauk SSSR, t. 109, 1956, p. 1098-1101 (in Russian). Zbl0131.31803MR83088
  29. [OKC] O.A. Oleinik, A.S. Kalashnikov, Y.L. Czhou, The Cauchy problem and boundary problems for equations of the type of nonstationary filtration, lzv. Akad. Nauk SSSR Ser. Mat., t. 22, 1958, p. 667-704 (in Russian). MR99834
  30. [V1] J.L. Vazquez, Asymptotic behaviour and propagation properties of the one-dimensional flow of a gas in a porous medium, Trans. Amer. Math. Soc., t. 277, 1983, p. 505-527. Zbl0528.76096MR694373
  31. [V2] J.L. Vazquez, Behaviour of the velocity of one-dimensional flows in porous media, Trans. Amer. Math. Soc., t. 286, 1984, p. 787-802. Zbl0524.35014MR760987
  32. [V3] J.L. Vazquez, The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc., t. 285, 1984, p. 717-737. Zbl0524.35060MR752500

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.